Benjamin Wood
Math 007
Professor Winkler
15 May 2013
Historically Significant Puzzles and Problems
There has been an assortment of mathematical puzzles and problems that
has had a great impact on history, for a variety of reasons. The effect of these
puzzles ranges from the development of probability, to the invention of graph
theory, to the determination of the value of pi. My fascination with historical puzzles
and problems lies in that they pull my love of history into the field of puzzles, and,
through these puzzles’ solutions, one can examine the historical impact that they
had on the world.
To qualify for this category, the puzzle or problem must be relatively old, for
a simple reason. Historical significance and import necessitates age because the
puzzle’s effects must be measurable and quantifiable today. One must be able to see
just how far reaching its effects were, and analyze it from an objective point of view.
Objectivity and comprehensive analysis demands the time for history to play out in
completion. Generally, this category also necessitates that the puzzle has a definite
solution, because it is often these solutions that have led to great changes and
revolutions in ways of thought. That being said, some of the most provocative and
revealing puzzles are those still bereft of solutions.
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The first of these
provocative puzzles is
Ostomachion by Archimedes,
circa 250 BC. Meaning literally
“bone-fight” in its Greek form,
presumably denoting what it
was made from, Ostomachion is
the basis for “tile” games and
initiated a great deal of
research into the field of combinatorics. In this “game,” there are fourteen unique
pieces arranged in a square.
The goal of the original game was to find out how many different arrangements of
the fourteen pieces can be made that still form a square of this size. This puzzle
embodies combinatorics, or the study of finite discrete structures, because that
exactly is the question at hand. Often considered the first true combinatorics puzzle,
it was not until 2003 that a definite solution for Ostomachion was arrived at. This
solution came at the hands of Bill Cutler, who is now a retired mathematician and
systems analyst. He proved, through the use of fractional area analysis, that there
are 536 distinct, non-reflected solutions, and when considering reflections of pieces,
there are 17,152 solutions. The 536 distinct ones are shown below, with the
highlighted text representing the most often seen representation of the puzzle (the
image above). The puzzle was finally solved when the area of the puzzle was
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analyzed and divided so that each piece contained a multiple of 1/48 of the total
area. I personally never would have thought to do this, but that is why I am not an
expert mathematician. It takes a special sort of mind to view a puzzle from a new
angle and understand what to do from there.
From this starting point, the
distinct fractional pieces could be
arranged in different ways to come
to the final answer. While this
solution is surely fascinating, the
problem itself is more important in
the overall metaphorical picture. In
this case, the problem itself is more
important in a historical context
simply because it is one of the first
mathematical puzzles on record. It
has remained a subject of interest
throughout history, as is evident
from its translation into many
languages, including German and ancient Arabic, from which our modern
understanding of it arises. That we still know what this puzzle is over two millennia
later is a testament to its importance in history.
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A millennia and a half later, we come to one of the first examples of
geometric growth: the famous Sissa’s Reward problem. Often known as the wheat
on the chessboard problem, this problem was apparently first described by Ibn
Khallikan in 1256 AD, and reads as follows: there was a Persian King who offered
Sessa a reward of his choice for inventing the chessboard, and Sessa requested one
grain of wheat on the first tile of the chessboard, two on the second, four on the
third, and so on for all 64 tiles. The King, not knowing any better, agreed, and ended
up owing Sessa 18,446,774,073,709,551,615 grains of wheat. To put this number in
perspective, this amount of wheat would weigh approximately 4x10^14 kilograms.
A simpler way to view this solution is
This puzzle’s significance lies in its contribution to the early study of
geometric growth. Its answer is far higher than most would intuitively expect, even
if they were expecting a large number. It must be noted that the date 1256 AD is not
definitive, and some estimates place this tale as first appearing earlier, closer to the
turn of the millennia in the epic poem Shahnameh. Ramifications of this problem
extend beyond just introducing graph theory, also serving to introduce the concepts
of exponents and zero powers. From this problem also stemmed an interesting
expression based on exponential growth. The “second half of the chessboard”
phrase refers to the point where it becomes economically unfeasible to continue
doing something in a business. This makes sense in the context of this puzzle
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because the second half of the chessboard contains more than two billion times as
much as the first half, and so its impact is much larger in the overall outcome. While
a coined phrase itself is trivial, its existence and use is suggestive of the influence of
the problem, and the weight it bears on the historical stage.
Next is one of the most famous and far-reaching mathematical puzzles in
history, the Konigsberg Bridges puzzle. The Konigsberg Bridges involves two sides
of a river and two islands:
It questions whether, given this
formation, it is possible to cross
all seven bridges once while at
the same time returning to the
starting point. It turns out that
this is not possible, and this was
proven to be so in 1736 by the
legendary Leonhard Euler while he was working in St. Petersburg. It is not possible
to travel across all bridges just once and end up at the start because each island or
shore must have only two or four bridges, with sets of two needed for the trip there
and complementary return. In this case, each has three, and so therein lies the
failing. Interestingly, solutions to these types of problems are today known as
Eulerian Paths in his honor. This puzzle is so exceptionally famous because it
inspired Euler to develop a great deal of what we know today as graph theory.
Graph theory is the study of the relationships and connections between different
locations, or “points” on a graph. The correlation between the Konigsberg Bridges
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problem and graph theory can be seen when the combination of land features and
bridges are viewed as points on a graph field that then are connected. Graph theory
today is used heavily in mathematics, computer science, and to model many
different types of physical systems and flows. Solely for the development of graph
theory, the Konigsberg Bridges problem will go down as one of the most significant
mathematical puzzles of history. Its ramifications are very far reaching when you
consider how essential graph theory is to things we take for granted every day, like
computer programming. As computer-driven technology becomes even more
prevalent than it is today, the importance of the Konigsberg Bridges problem will
grow with it.
Another puzzle that, without a doubt, falls into the historically significant
category is the “15 puzzle,” as it is commonly known. It states that there is a four by
four square of slide-able tiles, numbered 1-15 in order (from left to right and top to
bottom), with the final slot being empty to allow the movement of the rest, one at a
time. The interesting question posed
was whether or not it was possible to
return to this position, if you started
from the perfectly ordered set as seen
to the right, but with the 14 and 15 tiles
being in reversed places. Sam Lloyd
devised this task, with a $1000 reward
for whoever could solve it. This whole
situation is thought provoking because Sam Lloyd was not the inventor, even though
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he claimed to be. The true inventor was a postmaster by the name of Noyes Palmer
Chapman, who came up with it in 1874. Sam Lloyd further discredited himself by
posing this task, because it essentially was a publicity stunt to try to win him
support as the “true inventor.” It was discrediting because switching the 14 and 15
was proven impossible over a decade before he posed this question. This conflict of
ownership is intriguing because it sheds light on the world of mathematical puzzles,
and how much they meant even one hundred odd years ago. It does not truly matter
who invented the puzzle at the end of the day, because what matters is the puzzle’s
effect on math and history as a whole, but it is nonetheless interesting. This puzzle
proved to be important because it paved the way for the study of permutations and
the algorithms of moves. Additionally, this system is exactly the same for the puzzle
where one must “fix” a sliding jig that contains a picture in it. As a child, I did
hundreds of these sliding puzzles, except I did 5x5 versions, to the point where I
knew the algorithm to solve them no matter what they looked like. What is
engrossing, however, is that there are situations among these boards that are truly
impossible to solve or create from a previously solved board; the 14-15 switch, for
example. Interestingly, chess master and savant Bobby Fischer was exceptionally
talented at solving these sliding puzzles, and was known to do so in about 30
seconds.
The Rubik’s Cube will go down as one of the most significant mathematical
puzzles in history due largely to its addictive quality and incredible popularity. It
has affected entire generations and there are few in the developed world who do not
know what one is. With estimates today ranging well over 300 million sold
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worldwide, this is no surprise. Invented in 1974 by Ernö Rubik, it took off in 80s and
is still massively popular today. The effect of the Rubik’s Cube extends beyond the
direct affect the cube itself had, due to its massive popularity and associations with
puzzles of all sorts. Through its popularity, the Rubik’s Cube invited a great increase
in attention into the realm of mathematical puzzles, which is not something that
happens very often. I learned how to solve a Rubik’s Cube in high school because I
found it so fascinating, and thought it was deeply impressive to watch people solve
them. The cube itself consists of 3 x 3 x 3 smaller cubes that are colored so that,
when correctly configured, all sides are a solid color. As such, there are six distinct
colors. All told, there are 43,252,003,274,489,856,000 different arrangements, yet
only one initial configuration. The Rubik’s cube is important mathematically
because it brought group math to the fore, along with the processes that make up
algorithms. I personally learned how to solve a Rubik’s cube using algorithms, not
brute trial and error and comprehensive understanding, which is something that I
wish I had. The Rubik’s cube’s interesting nature arises because, following simple
algorithms, all 43 quintillion configurations can be returned to the one original
arrangement, and in a short period of time as well.
After discussing these problems and their significance at length, it is equally
important to consider what separates them from the rest. What separates them
from the average puzzle that is made today, and will be forgotten in five or ten
years? To this answer I respond that a puzzle must have several qualities in
particular, yet still at the end of the day, who can say for sure? There is an element
of luck to puzzle preservation that goes along with history itself.
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There is a common theme to the puzzles that I have grouped into the
category of historically significant, and it is this theme that causes a puzzle to last
through time. This theme is the physicality of each of these problems. Each one,
barring the Konigsberg Bridges problem, can be created or bought and played with
at ones leisure. Even the Konigsberg Bridges problem can just be drawn on a piece
of paper. For many, myself included, the best way to learn and think is hands-on.
The ability to manipulate a puzzle physically lends much to its appeal and life
expectancy because it appeals to a larger crowd. Additionally, the “15 puzzle,”
Konigsberg puzzle, Ostomachion, and the Rubik’s Cube can all be solved (or partially
solved in the case of Ostomachion) without any formal education. Their simplicity
and basis in logic and comprehensive thinking allows their solutions to be found
through thought alone.
Math knowledge, however, plays a definitive roll in solving them and
analyzing them more completely. For problems like Ostomachion and the Rubik’s
Cube, combinatorics is necessary to finding the total number of solutions or
variations. This application of math is another facet that adds depth to these
problems, and through depth, prolongs their lives.
I say that chance is involved because there is always the chance that a puzzle
will get skipped over; it might be brilliant, but not appreciated. Puzzles thrive on
popularity. To gain mainstream fame and prominence, a puzzle must extend beyond
outside of the confines of the puzzle community to reach the greater population.
There is always the chance that a puzzle will die before its adolescence for a variety
of reasons, like if the creator dies, or less drastically but of equal importance, if the
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puzzle originated in an uncommon language or place. To earn a place on the
historical podium a puzzle must have the characteristics above, and it must enchant
all types of people, but it also must by chance originate among the best of
circumstances.
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Works Cited
Bell, Tim. "Königsberg Bridge Problem." The Konigsberg Bridge Problem.
University of Caterbury, 3 Nov. 1999. Web. 16 May 2013.
"Combinatorics." Wikipedia. Wikimedia Foundation, Inc., n.d. Web. 16 May
2013.
"Graph theory." Wikipedia. Wikimedia Foundation, Inc., n.d. Web. 16 May
2013.
"Ostomachion." Wikipedia. Wikimedia Foundation, Inc., n.d. Web. 16 May
2013.
O'Connor, J. J., and E. F. Robertson. "Mathematical Games and Recreations."
Mathematical Games. St. Andrews, May 1996. Web. 16 May 2013.
Pickover, Cliff. "Ten of the Greatest: Maths Puzzles." Mail Online. DailyMail, 18
June 2010. Web. 16 May 2013.
Pitici, Mircia. "Archimedes’ Stomachion." Untitled Document. University of
Cornell, Sept. 2008. Web. 16 May 2013.